It follows from (REF ) and (REF ) that \(e^{\gamma _ku_j^{N/(N-1)}}\)  is bounded in \(L^r(\mathbb {B})\)  for some \(r>1\) ,
and thus \(e^{\gamma _ku_j^{N/(N-1)}}\rightarrow e^{\gamma _ku_k^{N/(N-1)}}\)  in \(L^1(\mathbb {B})\)  as \(j\rightarrow \infty \) . Hence
\(\int _\mathbb {B}e^{\gamma _ku_k^{N/(N-1)}}dx=\Lambda _{\gamma _k,\alpha }\)  and \(u_k\)  is the desired extremal function.
Clearly \(\Vert u_k\Vert _{1,\alpha }=1\) . Moreover, the Euler-Lagrange equation of \(u_k\)  is (REF ). According to the regularity theory for degenerate elliptic equations, see Serrin [1]}, Tolksdorf [2]} and Lieberman [3]}, we have
\(u_k\in C^1(\overline{\mathbb {B}})\) . \(\hfill \Box \)
